The Latics welcome David Moyes' side to Boundary Park on Saturday, hoping to pull off a second successive Merseyside shock after being Liverpool.
Royle told evertontv: "I think this current Everton side is somewhat immune from the kind of performance Liverpool had, albeit not with their strongest side. Everton are now a very solid outfit. They don't lose many games. If David plays his strongest team, I don't think we can lose this game.
"But I'm delighted Oldham are in the round quite honestly. They beat our neighbours in a game they weren't supposed to win. The whole town is still very excited.
"I've got sons, a daughter-in-law and grandchildren all living in Oldham. I get across there and believe me, the place is buzzing. I've been having requests for tickets from this side and the opposite side so I'm using all my allocation."